Stirling’s formula gives an approximation for $n!$, the factorial function. It is

We can derive this from the gamma function. Note that for large $x$,

where

with $0<\theta<1$. Taking $x=n$ and multiplying by $n$, we have

Taking the approximation for large $n$ gives us Stirling’s formula.

There is also a big-O notation version of Stirling’s approximation:

We can prove this equality starting from (2). It is clear that the big-O portion of (3) must come from $e^{{\frac{\theta}{12n}}}$, so we must consider the asymptotic behavior of $e$.

First we observe that the Taylor series for $e^{x}$ is

But in our case we have $e$ to a vanishing exponent. Note that if we vary $x$ as $\frac{1}{n}$, we have as $n\longrightarrow\infty$

We can then (almost) directly plug this in to (2) to get (3) (note that the factor of 12 gets absorbed by the big-O notation.)